Adam Clarke on 1 Timothy 1:2

My own son in the faith - Brought to salvation through Christ by my ministry alone. Probably the apostle speaks here according to this Jewish maxim: כל המלמר בן תכירו תורה מעלה עליו הכתוב כאלו ילרו He who teaches the law to his neighbour's son is considered by the Scripture as if he had begotten him; Sanhedrin, fol. xix. 2. And they quote Num 3:1, as proving it: These are the generations of Aaron and Moses - and these are the names of the sons of Aaron. "Aaron," say they, "begot them, but Moses instructed them; therefore they are called by his name." See Schoettgen.

But γνησιῳ τεκνῳ may mean my beloved son; for in this sense το γνησιον is not unfrequently used.

In the faith - The word πιστις, faith, is taken here for the whole of the Christian religion, faith in Christ being its essential characteristic.

Grace, mercy, and peace -

Grace, the favor and approbation of God.

Mercy, springing from that grace, pardoning, purifying, and supporting.

Peace, the consequence of this manifested mercy, peace of conscience, and peace with God; producing internal happiness, quietness, and assurance.